T2 



Geodesic Lines on the Syntractrix of Revolution. 



E. L. Hancock. 



The syntractrix is defined as a curve formed by taking a constant 

 length, d npon the tangent c to the tractrix*. Tlie surface formed by re- 

 volving this curve about its asymptote is the one under consideration. We 

 shall call it S. 



Being a surface of revolution it is represented by the equations 

 x^u cos V 

 y = u sin v 



z = — v/d2 — uM-§log 



2 '^d — v/d2 — u2 

 Using the Gaussian notationf we find: 



^^ u^(d^-2cd)+cM2 ^^ ^^^, __^I^g^^^^^ B = - 



U2(d2 — U2) ' lVd2 — U2 



Ti" — cd . u2(d2 — 2cd)+cd3 -r,, ^ -n./ u(u2 — cd) 



7== U Sin V C = U, D r=r } L^ , D ^ O, D ' =: 7^= 



U/(i2_u2 U(d2 — U2)| l/d2— U2 



_ 1 _ DW—W^ _ (u2 — cd)[u2(d — 2c) +cd2] 

 ^ = R^~ EG — F2 — (d2 — u2)[u2(d — 2c)-t-c2d 



In tlie particular surface given by d=:2c the Gaussian ciu-vature be- 

 comes 9. o d2i 



d2 — U2 



Here d is positive, and since d > u, the denominator is always positive. 

 We get tlie character of the curvature of ditferent parts of the surface by 

 considering the numerator. When u2= d2 2, K = 0, i. e., the circle u = d/2 



-y=: is made up of points having zero-curvature. When u2 > d2/2, K> O, 



and when u2 < d2/2, K < O. 

 For tliis particular sui'face 



,14 • 2u2--d2 



E = ~ , F = o, G = u2, A = — ^-v==„ u cos v, B = — 



4u2(d2_u2) 2u-/(i2 — U2 



2u2 — d^ i\\ u(2u- — d2 

 usinvC = o, D=^ -^ ._, D' = o, D'^=^ 



2u/d^Zr^ " ^'" > V.-U, ^_— ^__^-, ^ _., 2/0^3.-^ 



To get tlie geodesic lines of the surface we make use of the method of 

 the calculus of variations according Weierstrass§. This requires that we 

 minimize the integral : 



'" Peacock, p. 175. 



t Bianehi, Differential Geometric, pp. 61, 87, 105. 



l Osgood, Annals of Mathematics, A'ol. II (1901), p. 105. 



